A Characterization Theorem for the Distribution of a Continuous Local Martingale and Related Limit Theorems

TL;DR

This paper establishes a characterization theorem for the distribution of continuous local martingales with absolutely continuous quadratic characteristics and derives related functional limit theorems.

Contribution

It introduces a new characterization theorem linking the distribution of local martingales to their quadratic characteristics and proves associated functional limit theorems.

Findings

Distribution of local martingales is determined by quadratic characteristics.

Functional limit theorems are established based on this characterization.

Provides a framework for analyzing continuous local martingales.

Abstract

The main result of the article reads: the distribution of a continuous starting from zero local martingale whose quadratic characteristic is almost surely absolutely continuous with respect to some non-random increasing continuous function is determined by the distribution of the quadratic characteristic. Functional limit theorem based on this characterization are proved.